A general method of Bayesian estimation for parametric imaging of the brain

We report a general method of Bayesian estimation that uses prior measurements to improve the signal-to-noise ratio of parametric images computed from dynamic PET scanning. In our method, the ordinary weighted least squares cost function is augmented by a penalty term to yield Φ(K,S)=minK{(C−f(K))TΩC−1(C−f(K))+SΦ(K,S=0)(K−K^)TΩK−1(K−K^)}, where C is a PET concentration history and ΩC is its variance, f is the model of the concentration history, K = [k1,k2,…,km]T is the parameter vector, K^ is the vector of population means for the model parameters, ΩK is its covariance, ΦK(K,S = 0) is the conventional weighted sum of squares. S > 0 is chosen to control the balance between the prior and new data. Data from a prior population of subjects are analyzed with standard methods to provide maps of the mean parameter values and their variances. As an example of this approach we used the dynamic image data of 10 normal subjects who had previously been studied with 11C-raclopride to estimate the prior distribution. The dynamic data were transformed to stereotactic coordinates and analyzed by standard methods. The resulting parametric maps were used to compute the voxel-wise sample statistics. Then the cohort of priors was analyzed as a function of S, using nonlinear least squares estimation and the cost function shown above. As S is increased the standard error in estimating BP in single subjects was substantially reduced allowing measurement in BP in thalamus, cortex, brain stem, etc. Additional studies demonstrate that a range of S values exist for which the bias is not excessive, even when parameter values differ markedly from the sample mean. This method can be used with any kinetic model so long as it is possible to compute a map of a priori mean parameters and their variances.